# A diagonal operation on power series

Given a formal power series $f(x,y)=\sum_{n,m\ge 0}f(n,m)\: x^n y^m$ in two variables $x$ and $y$ over some field of characteristic zero, e.g. the field of complex numbers $\mathbb C$, we define a new formal power series in one variable $t$:

$\Delta(f)(t):=\sum_{i\ge 0} f(i,i) \:t^i$.

It is known that if $f(x,y)$ is rational $\Delta(f)(t)$ is in general not rational (I think it is algebraic though).

Example: $f(x,y)=\frac{1}{1-x-y}$ leads to $\Delta(f)(t)=(1-4t)^{-1/2}$.

The question is: Is there a constructive way (i.e. algorithm or explicit formula) to calculate $\Delta(f)(t)$ for rational (or algebraic) $f(x,y)$ ?

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When you say "it is known that", do you have a reference? –  Igor Rivin Aug 16 '12 at 15:01
That is, you give an example (which works for me) but what book do you know which discusses these matters? –  Igor Rivin Aug 16 '12 at 15:02
What about $\Delta(f)(t)=\mathrm{Res}_{z=0} f(tz,1/z)/z$ –  Pietro Majer Aug 16 '12 at 15:03
@Pietro What you wrote is a really good mind-boggling exercise for the students: if your formula were correct, $\Delta f$ would be rational for rational $f$ (computing the residue of a rational function requires a few differentiations only and they cannot kill the rational dependence on the parameter $t$). It took me 3 full minutes to discover the error, so I do not want to deprive the others from the pleasure of figuring it out by themselves :). –  fedja Aug 16 '12 at 15:27
Nice! Actually I wasn't thinking to a particular context, that in any case has to be fixed (even though I'd had bet rational functions were ok!) –  Pietro Majer Aug 16 '12 at 16:38
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Yes, there is a constructive way for both rational and algebraic $f.$ You should check out the very nice paper by Adamszewski and Bell, and references there in -- the formula for rational functions is given by Deligne (ref [13] in the cited paper), the result for algebraic functions appears very deep (see papers by Andre and Christol cited in the reference), but anyway, just read the introduction to the paper.